B = bessel(alpha,x)
is called Bessel's equation, and its solutions are known as Bessel functions.
B = bessel(alpha x)
computes Bessel functions of the first kind, or modified Bessel functions of the first kind, for real, nonnegative alpha
and nonnegative x
:
x
(no imaginary component), bessel
uses the besselj
function to compute the Bessel function of the first kind.x
(no real component), bessel
uses the besseli
function to compute the modified Bessel function of the first kind.x
(both real and imaginary component), bessel
uses the bessela
function to compute the Bessel function of the first kind.alpha
is a scalar and x
is a vector, B
is a vector the same length as x
. If x
is a vector of length m
and alpha
is a vector of length n
, then B
is an m
-by-n
matrix and B(i,k)
is bessel(alpha(k), x(i))
. The elements of x
can be any nonnegative values, in any order. For alpha
, the increment between elements must be 1, and all elements must be between 0 and 1000, inclusive.
x = 0:.25:10;
plot(x,[bessel(0,x);bessel(1,x)])
x
, bessel
calls the function besselj
. besselj
evaluates the Bessel function using three-term recurrence for small-valued x
, or an asymptotic series for large x
. For purely complex x
, bessel
calls besseli
, which uses three-term backward recurrence for most x
, and an asymptotic expansion for large x
. For complex x
, bessel
calls bessela
, which uses either a power series or an asymptotic expansion depending on roundoff error considerations. See reference [1].
For some values of arguments, bessela
's results may be severely contaminated by roundoff error. [J,digits] = bessela(alpha,x)
returns an estimate of the number of correct significant digits in the computed result. digits
is the log10
of the estimated relative error, so a value of 14 or 15 corresponds to nearly full accuracy in IEEE or VAX arithmetic, while 1 or 2 indicates nearly useless results. Any negative value of digits is replaced by zero, the corresponding J
set to NaN
and a division by zero warning message is generated. If either alpha
or x
is less than 50, digits are at least 8. In the (alpha,x)
plane, the region of least accuracy is near the line alpha = x
, so small values of alpha
and large values of x
, or vice versa, give the most accurate results. For example:
---------------- x alpha digits ----------------25
25
11.825
75
14.475
25
14.575
75
1.9 ----------------
besseli
,besselj
,besselk
,bessely
(c) Copyright 1994 by The MathWorks, Inc.